# Fluid passing through a convergent-divergent nozzle

#### T C

Summary
I want to know what can be possible output velocity when pressurised fluid passes through a convergent-divergent nozzle.
Suppose, there is a convergent-divergent nozzle where the inlet and the outlet both has 5 sq cm area while the throat has 1 sq cm area. A pressurised compressible fluid at 5 barA pressure having velocity 10 m/s entered the c/d nozzle. The back pressure i.e. the pressure at the outlet is 1 barA pressure. I want to know what can be possible velocity of the fluid at the outlet. My common sense and little understanding of nozzle properties tells me that the velocity is around 250 m/s.

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#### russ_watters

Mentor
A pressurised compressible fluid...
What fluid? Yes, it matters.
My common sense and little understanding of nozzle properties tells me that the velocity is around 250 m/s.
Are you saying you calculated this or just guessed? If you calculated it, how?

#### cjl

A fluid exiting a converging-diverging nozzle with an area ratio of 5 will be traveling Mach 3.17. How fast this is in m/s depends on which fluid it is, and what the temperature of the fluid is. However, unfortunately for your setup, the exit pressure with an area ratio of 5 is just over 2% of the reservoir pressure, so with only 5 bar in the reservoir and 1 bar ambient pressure, you're likely going to have flow separation and probably even a normal shock in the exit. If you want smooth flow through this nozzle with a supersonic exit, you'll need at least 25 or 30 bar in the reservoir to avoid this problem (and ideally 50 bar).

What's the application for this, and how did you come to the 250m/s conclusion?

#### T C

This way: In the convergent section, the flow at the throat will be five times that of the inlet and at the exit, the flow will be 5 times that of the throat.

#### cjl

Why would you expect that to be the case?

#### T C

Conversion of enthalpy into velocity.

#### cjl

That just tells you that the fluid will speed up. It doesn't tell you by how much. Have you done anything with compressible flow before?

#### russ_watters

Mentor
Also, there appears to me to be a problem with the setup where the states are over-constrained and the given parameters incompatible: the inlet velocity is impossibly slow for such a large pressure ratio (this is related to what cjl was saying.

You seem to be treating this kind of like a supersonic flow nozzle, without the supersonic flow. The behaviors are very different.

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#### cjl

Also, there appears to me to be a problem with the setup where the states are over-constrained and the given parameters incompatible: the inlet velocity is impossibly slow for such a large pressure ratio (this is related to what cjl was saying.
While the inlet velocity is a bit slow, I'm not really concerned about that. 5:1 convergent section on a C-D nozzle is enough that the inlet velocity is only around mach 0.1 or so, and that's close enough to just a static condition that I'm not worried about the exact conditions there. Given the pressure ratio, the flow will choke in the throat, and the upstream parameters will change a bit because of this, but it won't really change the flow much if the actual inlet velocity at the entrance to the convergent section ends up being 20 or 30m/s rather than 10. The bigger problem is that the exit conditions aren't reasonable for a nozzle like this - you'd need a much smaller expansion ratio if you wanted to keep the 5:1 pressure ratio (around 1.35:1), or a much larger pressure ratio if you wanted to keep the 5:1 area ratio (around 50:1, as I said). That's why I'm curious what the background and application for this question is, so I can direct OP in a productive direction.

#### russ_watters

Mentor
While the inlet velocity is a bit slow, I'm not really concerned about that. 5:1 convergent section on a C-D nozzle is enough that the inlet velocity is only around mach 0.1 or so, and that's close enough to just a static condition that I'm not worried about the exact conditions there. Given the pressure ratio, the flow will choke in the throat, and the upstream parameters will change a bit because of this, but it won't really change the flow much if the actual inlet velocity at the entrance to the convergent section ends up being 20 or 30m/s rather than 10. The bigger problem is that the exit conditions aren't reasonable for a nozzle like this - you'd need a much smaller expansion ratio if you wanted to keep the 5:1 pressure ratio (around 1.35:1), or a much larger pressure ratio if you wanted to keep the 5:1 area ratio (around 50:1, as I said). That's why I'm curious what the background and application for this question is, so I can direct OP in a productive direction.
Well, given that the OP "calculated" an outlet speed that's subsonic for air, my concern is he may believe he can get that velocity increase with purely subsonic flow. Either way, he'll need to clarify exactly what he's trying to do and which are the real and guessed inputs/constraints and outputs/results.

cjl

#### cjl

Yeah, I don't want to speculate too much on the actual application here, but there definitely needs to be more clarity here.

#### T C

Kindly tell me what can be the possible velocity at the throat and how that can be calculated.

Gold Member
A fluid exiting a converging-diverging nozzle with an area ratio of 5 will be traveling Mach 3.17. How fast this is in m/s depends on which fluid it is, and what the temperature of the fluid is. However, unfortunately for your setup, the exit pressure with an area ratio of 5 is just over 2% of the reservoir pressure, so with only 5 bar in the reservoir and 1 bar ambient pressure, you're likely going to have flow separation and probably even a normal shock in the exit. If you want smooth flow through this nozzle with a supersonic exit, you'll need at least 25 or 30 bar in the reservoir to avoid this problem (and ideally 50 bar).

What's the application for this, and how did you come to the 250m/s conclusion?
Actually, for the given pressure ratio, there should be no normal shocks in the nozzle. You need $p_b/p_{01} \leq 0.24$ to set up a flow with no normal shocks for this expansion ratio. Unfortunately for @T C, the requirement for no shocks at the exit whatsoever is for that ratio to be 0.02, so this is an overexpanded nozzle (and there will be oblique shocks at the exit. They will likely be nearly normal, too, since the pressure ratio is so close to that of a normal shock at the exit.

Kindly tell me what can be the possible velocity at the throat and how that can be calculated.
This is challenging since you appear to have zero background in compressible flow, particularly based on your previous similar threads (below). The bottom line here is that the maximum attainable velocity from a given reservoir pressure is determined by the total enthalpy of that reservoir. If you convert 100% of that enthalpy into velocity, that corresponds to the maximum attainable velocity. For a given reservoir condition and nozzle geometry, one simply must increase the reservoir temperature to increase the output velocity.

cjl

#### russ_watters

Mentor
Kindly tell me what can be the possible velocity at the throat and how that can be calculated.
We can't. You need to fully define the problem in order for it to be solvable:
-What fluid is it?
-What is the temperature?
-Which of the inputs - pressure or velocity - is the one you really want?
-What is your actual goal (so we can correct the problem if necessary)?

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#### T C

This is challenging since you appear to have zero background in compressible flow, particularly based on your previous similar threads (below). The bottom line here is that the maximum attainable velocity from a given reservoir pressure is determined by the total enthalpy of that reservoir. If you convert 100% of that enthalpy into velocity, that corresponds to the maximum attainable velocity. For a given reservoir condition and nozzle geometry, one simply must increase the reservoir temperature to increase the output velocity.
The "total enthalpy" is a complicated factor here as when all enthalpy is squeezed out, the temperature will reach absolute zero. What I want to know is what's the maximum attainable velocity with the specific nozzle that I have mentioned.
-What fluid is it?
Consider it to be saturated steam.
-What is the temperature?
143.64°C i.e. the boiling point of steam at 5 barA pressure.
-Which of the inputs - pressure or velocity - is the one you really want?
Velocity.

#### russ_watters

Mentor
Consider it to be saturated steam.
143.64°C i.e. the boiling point of steam at 5 barA pressure.
Velocity.
In that case you have a fully subsonic and incompressible flow situation - a Venturi Tube - and the outlet is exactly the same as the inlet: 10m/s, 5bar, saturated steam.

Note, since you declared these to be the actual constraints, you've discarded the 1 bar outlet pressure constraint.

#### T C

Note, since you declared these to be the actual constraints, you've discarded the 1 bar outlet pressure constraint.
How?

#### russ_watters

Mentor
A system has a characteristic performance curve. At a point in the system, a specified pressure (differential or pressure ratio vs the outlet) is associated with one and only one flow rate. You can't declare that you want both a certain differential pressure and a certain flow rate. Fixing the inlet flow rate to be very slow means the differential pressure (the pressure that drives the fluid through the system) is roughly zero. Hence, the inlet and outlet pressure are the same.

#### T C

I have never said that I want to have a specific flow rate. I just want to know what can be the possible flow rate with the criteria. Kindly don't put words in my mouth.

#### russ_watters

Mentor
I have never said that I want to have a specific flow rate. I just want to know what can be the possible flow rate with the criteria. Kindly don't put words in my mouth.
Yes you did: you said you want 10m/s at a 5 bar into 5 sq cm, which fixes the flow rate. I did mix terminology there a bit (volumetric or mass flow rate vs flow velocity), but given your constraints it didn't actually matter. And not for nothing, but you just did too: your OP says velocity and you just said flow rate..

And drop the attitude: this isn't an argument, I'm trying to help -- we all are.

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#### T C

Now I can point out my fault. I have forgot that if the velocity is <Mach 1, in that case divergent nozzles convert velocity into enthalpy. So, I now want to change the topic a little bit and want to make it simple. What will happen if a compressible fluid (say saturated steam) at 5 barA pressure is released through a divergent nozzle of 5:1 ratio to atmospheric pressure. I can calculate what can be the possible velocity when steam (or any compressible fluid) is released from 5 barA to 1 barA pressure. What I want to know is whether the divergent nozzle can further convert more enthalpy into velocity or not.

#### cjl

If your reservoir pressure is 5 bar, and ambient is 1 bar, your flow will choke in the throat. It will then further accelerate in the divergent section, but a 5:1 nozzle requires a pressure ratio of 50:1 to operate optimally, so you'll end up with shocks and flow separation at the exit as the exit flow will be at only 0.1 bar but ambient pressure is 1 bar.

Making the flow saturated steam makes this even more complicated, as you'll get condensation in the flow as it expands through the nozzle and cools.

#### T C

If your reservoir pressure is 5 bar, and ambient is 1 bar, your flow will choke in the throat. It will then further accelerate in the divergent section, but a 5:1 nozzle requires a pressure ratio of 50:1 to operate optimally, so you'll end up with shocks and flow separation at the exit as the exit flow will be at only 0.1 bar but ambient pressure is 1 bar.
How you have calculated that the pressure at the exit will be just 0.1 barA.

#### cjl

There are some fairly well known (at least among aerodynamicists) relations telling you how a compressible, isentropic flow will behave in a converging or diverging duct. They can be found here:

In these relations, A* is the throat area, so if there's an expansion ratio of 5, A/A* is equal to 5. From that, using equation 9, you can find that for a nozzle with that expansion ratio, the mach number at exit is equal to 3.175. You can then plug this into equation 6 and discover that at mach 3.175, the pressure is equal to 0.021 times the total pressure. Total pressure is 5 bar, so exit pressure for the nozzle is 0.105 bar. You can also just plug the numbers into a calculator like this one, which is just using the equations above.

Also, as I said, this is only true for a gas flow. If you have saturated steam, this won't be accurate, since you'll get condensation in the nozzle leading to (effectively) heat addition in the flow, as well as droplets. At 5 bar, saturated steam is only at 155C or so, but if you feed a 5:1 C-D nozzle with 5 bar at 155c, the exit temperature is -131c. This would be fine if your fluid is air, or something similar, but if you want to use steam, you'll need a pretty significant amount of superheat to prevent condensation.

#### russ_watters

Mentor
So, I now want to change the topic a little bit and want to make it simple. What will happen if a compressible fluid (say saturated steam) at 5 barA pressure is released through a divergent nozzle of 5:1 ratio to atmospheric pressure. I can calculate what can be the possible velocity when steam (or any compressible fluid) is released from 5 barA to 1 barA pressure. What I want to know is whether the divergent nozzle can further convert more enthalpy into velocity or not.
If your reservoir pressure is 5 bar, and ambient is 1 bar, your flow will choke in the throat. It will then further accelerate in the divergent section....
Note that he said divergent only this time, so for this to be true you have to assume there's still/also a convergent section and a large enough reservoir feeding it. And the 5bar is the pressure at the throat, not the reservoir.

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